As most philosophers, I am a nervous wreck. And as a nervous wreck, I have struggled in the past to find ways to unwind that would enable me to plug off the intense demand for new ideas and the many tasks in need of fulfillment. Rather, I was looking to enjoy something far from philosophical research, enjoyable in itself and without the need for intricate philosophical reflection. As it turns out, when I started doing origami as a pastime, I realized that—on the contrary of what they say—if Muhammad won’t go to the mountain, the mountain will come to Muhammad.

Origami, the traditional Japanese art of paper folding, is a practice with a very long history. The oldest evidence of decorations made through folded paper in Japan, then known as origata, dates back to the XIV or XVth century, with the first book with instructions on how to fold them dating back to 1747. In the last century, origami gained international visibility, thanks to the work of many Japanese and international origami artists and the establishment of dozens of national and international associations and conventions. Needless to say, while it is intuitive to think of paper cranes or frogs, contemporary origami has reached unbelievable levels of detail and complexity.

[Fig. 1: An origami sea turtle, model designed by Jang Yong Ik. Folded and photographed by the author.]

I used to love making simple origami as a kid in my late elementary school, and found my way back into making them a couple of years ago. However, I only recently discovered a crucial aspect to origami one wouldn’t normally associate with either tradition, art, or fun: math. To design and make an origami, it is necessary to apply a sequence of folds on a piece of paper, which has typically a square shape: each crease corresponds to a geometrical operation, with physical limitations regarding how the paper can be folded based on the previous steps. In this sense, origami has been used to teach mathematics in kindergartens since the late 19th century. At the same time, mathematicians such as Robert J. Lang developed an algorithm capable of creating origami by identifying the crease pattern that, once folded, leads to an almost finalized model. The geometric complexity of these patterns can be astounding, but not surprising when looking at what creasing them leads to (see figure 2). The mathematical structure of origami has even contributed to the field of robotics: for example, some engineers developed robots capable of movement through self-folding, while other origami-inspired robots have potential biomedical applications.

[Fig. 2: an origami crane and its crease pattern. Folded and photographed by the author]

As someone who just sought to relax and channel my natural restlessness into nice things, the complex mathematics underlying origami was a wild discovery. I had managed to fold (although not design) quite complex models myself, some of which required 250-300 folds. Yet I found myself underappreciative of these fun little paper sculptures—not only due to their complexity, but to the logic characterizing even the simpler designs. I thought of liking origami as a child, and wondered whether I just liked the pretty sculpture I was able to make, or whether it was the methodical procedure of precise and simple folds through which those sculptures were made that I truly enjoyed. It was as if the elementary sequence of steps—combined in a rational sequence—was the source of the origami’s beauty.

Immanuel Kant famously argued, in his Critique of the Faculty of Judgement, that what makes certain things, either natural or man-made, beautiful is their apparent teleology, the impression that they are designed with some specific end. He proposes that the reason why people find things beautiful is that those things express the purposefulness of the world. To experience and judge a thing as beautiful, be it a flower, a landscape or a poem, is to view that thing as appearing to have “an end without end”: we judge something as beautiful because it appears, to our intellect and senses, as if it was made to please us. A beautiful thing is not necessarily made with the end of being beautiful, but we perceive it and, to some extent, understand it as if it was made for us, made to be experienced as pleasant to our minds and senses. According to Kant, there is a harmony, or even purposefulness, in our experience of beautiful things—not because there is indeed such a harmony intrinsic to the beautiful thing, but because the experience of beauty produces such harmony within us, and in our relation to the beautiful thing. This experience is constituted by what Kant calls a free play between our intellect and our faculty for imagination, a feeble balance between our rationality and our capacity to wonder. The harmony and free play produced in us (thanks to our experience of beauty) is what makes us judge it as beautiful.

I found myself wondering whether—back in my childhood—I had already found a kind of calm in the experience of the harmony that is needed to make origami. The easy origami children can make are often quite simple and not very realistic: even the universally recognizable origami crane is quite abstract, schematic, compared to the real thing. Even in the case of more complex origami, there are often evident regularities in their appearance, with frequent symmetries and regular shapes. At a first glance, a Kantian view of the beauty of origami might pick up on their formal regularities. Indeed, there is something truly fascinating in the reproduction of natural and man-made entities through such regularities, as a reassuring expression of rationality intrinsic to the paper sculpture and, by proxy, to what it represents.

However, when I make origami these days and think about what I find beautiful about them I cannot help but think about not just the finished model, but the very process that leads me there. After all, for Kant, the harmonious experience of beauty is not in its object, but in my relationship with and experience of it. The feel I developed for appreciating the different consistencies and thickness the paper can have; the capacity to interpret increasingly complex instructions, and to perform more precise and detailed folds; and the shift from wondering how does a paper square lead to a sea turtle, into feeling wonder by coming to understand the most crucial sequences to fold its body structure, its shell, its fins. These experiences I have when folding paper are not intrinsic to the origami as an object, but rather to my relationship with the origami as a practice, an experience un-folding through my folding (pun fully intended). The apparent purposefulness of the origami as a beautiful object—and a beautiful practice—comes from my appreciation of the formal geometry of the procedure and its path into a realistic model: indeed, the free play of rationality and imagination. If beauty results from such harmonious free play and appears as a sense of purposefulness and meaningfulness, my experience of beauty emerges from partaking in origami-making, as I seek to configure the paper into a model. 

It is somewhat funny that one of my ways out of philosophical work has led me right back to philosophy, through the conjunction of mathematical logic and art. While I wanted to move away from philosophy in folding paper, that is a somewhat soothing way back for me. And now, I can’t really help reflecting on my own thoughts and the balance of rationality and imagination when I fold—from paper, to algorithm, to sea turtle.

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